II. " On the Partitions of the r-Pyramid, being the first class 

 or r-gonous #-edra." By the Rev. T. P. KIRKMAN, M.A., 

 E.R.S. Received October 14, 1857. 



(Abstract.) 



Partitions proper of the r-pyramid are made by drawing diagonals 

 none crossing another in the r-gonal base, and diapeds (intersections 

 of non-contiguous faces) none enclosing a space, in the r-edral vertex. 

 The object of the memoir is to enumerate the number of such par- 

 titions that can be made with K diapeds in the vertex and k diagonals 

 in the bases of the pyramid. By the drawing of k diagonals, the 

 pyramid becomes a (r-J- l)-acral (r + k+ l)-edron, which by the in- 

 troduction of K diapeds becomes a (r-j-K + l)-acral (r-f- k+ l)-edron. 

 Such a figure is termed an r-gonous (r-f- K-f- l)-acral (r+k+ l)-edron 

 of the first class. The definition of an r-gonous #-edron of the first 

 class is that it contains a discrete r-gony, i. e. K diapeds and k 

 diagonals of which no diaped meets a diagonal, and such that the 

 convanescence of the K diapeds will form an r-ace, and the eva- 

 nescence of the diagonals forms an r-gon. 



If the summits upon the k diagonals be, one or more of them, 

 partitioned by K f diapeds, or the faces about the K diapeds be par- 

 titioned by k' diagonals, there arises a mixed r-gony, in which are one 

 or more angles made by a diaped and a diagonal. If such a figure 

 has not a discrete r-gony as well as that mixed one, and has no 

 (r_f-/.')-gony, by the vanescence of which the (r-fr') -pyramid can 

 be obtained, it is an r-gonous ^r-edron of the second class. And 

 r-gonous #-edra of the third class can be obtained by partitioning 

 the faces about the K' diapeds and the summits upon the k' diagonals, 

 in such a manner that no (r-fr')-gony shall be introduced ; and so 

 on for higher classes of r-gonous ,r-edra. 



It is proved that every partition proper of the r-pyramid, that is, 

 any (l+'K) -partitioned r-ace laid on a (1 + k) -partitioned r-gon, is 

 an r-gonous (r+ k+ l)-acral (r-f k -f- 1 )-edron. The number of the 

 (1 +k) -partitions of the r-gon, and of the (1 +K) -partitions of the 

 r-ace is known by the formulae given in the author's memoir " On 

 the partitions of the r-gon and r-ace," in the Philosophical Trans- 

 actions, 1857. The present memoir gives the formulae whereby the 



